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Irreducible components of varieties of representations of a class of finite dimensional algebras

Marko Čmrlec

Abstract

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that the subvarieties with fixed radical or socle layering are irreducible. We then compare the coverings we get using radical and socle layerings to determine the components.

Irreducible components of varieties of representations of a class of finite dimensional algebras

Abstract

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that the subvarieties with fixed radical or socle layering are irreducible. We then compare the coverings we get using radical and socle layerings to determine the components.

Paper Structure

This paper contains 5 sections, 23 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Irreducibility
  4. Possible radical layerings
  5. Containments

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field with $\operatorname{char}(k)\neq 2$, and $\mathcal{A}=k\langle x_1,\dots,x_n\rangle/I$ the algebra of the algebra of noncommutative polynomials, quotiented by the ideal where $S=\sum_{i,j=1}^n a_{ij}x_ix_j$ with $\det(a_{ij})\neq 0$. Let $d\in\mathbb{N}$. Then $\mathop{\mathrm{rep}}\nolimits_d \mathcal{A}$ admits a decomposition into irreducible components

Figures (1)

  • Figure 1: Set of roots of $q$ and the generators of its $\mathbb{N}$ linear closure. for $3$-tuples and $4$-tuples of matrices. The dimension vector that does not appear by lemma \ref{['izjemna upodobitev']} is marked in red.

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof : Proof of lemma \ref{['lema_odp']}
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • ...and 40 more